【作者】 李慧民；

【导师】 杨晓松；

【作者基本信息】 华中科技大学 ， 系统分析与集成， 2007， 博士

【摘要】 切换系统是一类由若干个连续或离散子系统或连续和离散子系统和一个作用在其上的切换法则组成的混合系统。它有着广泛的实际背景和复杂的动态行为,从而引起了国内外学者的普遍关注。BZ化学反应为目前非线性化学动力学中最重要的试验研究和理论分析对象之一,该反应以结构简单,非线性动力学行为丰富而受到人们的极大关注。在本文中,我们主要讨论含有离散子系统的切换系统的稳定性。主要研究内容概括如下:1.对一个一般的非自治非线性离散切换系统,我们利用多重Lyapunov函数和离散系统的比较原理给出了该离散切换系统双测度稳定的一个结果,当离散子系统的个数有限时,我们在此基础上又给出了一个更细致的结果。接着我们对带有外部输入的非线性离散切换系统的输入状态稳定、具有权重的输入状态稳定、具有权重的可和输入状态稳定进行了研究,利用平均驻留时间方法分别给出使这个离散切换系统输入状态稳定、具有权重的输入状态稳定、具有权重的可和输入状态稳定的充分条件。作为上述输入状态稳定结果的一个应用,离散串联系统全局渐近稳定的充分条件很容易被给出。2.对一类离散切换系统-串联离散切换系统,我们先利用Brower不动点定理给出了其存在周期轨道的充分条件,接着我们讨论周期轨道的稳定性,得到了周期轨道稳定的一个结果,此外,我们对这类离散切换系统的一个特殊情形-只有一个子系统的情形给出周期轨道最小正周期和个数的估计,并举例说明这些结果是无切换时独有的。3.对一类新切换系统-由连续和离散子系统组成的切换系统,我们首先利用多重Lyapunov函数、连续和离散系统的比较原理给出了这类切换系统双测度稳定的一个结果,当连续和离散子系统的个数都有限时,我们在这个结果的基础上给出了一个更细致的结果,其次利用平均驻留时间方法我们得到了这类切换系统全局渐近稳定的充分条件,最后对由线性时不变的连续和离散子系统组成的切换系统我们给出其在周期切换信号下全局渐近稳定性的一个结果。4.同样对这类新切换系统-由连续和离散子系统组成的切换系统,我们讨论了它的鲁棒稳定性。首先对这类切换系统在多胞型摄动下的稳定性进行了研究,获得其二次稳定的充要条件,最后对这类由连续和离散子系统组成的切换系统的一个非线性情形即离散子系统有非线性扰动的情形我们给出了其状态在任意切换律下的一个估计,并从这个状态估计中我们获得了其在任意切换律下全局渐近稳定的一个充分条件。此外,我们还研究了一个典型的BZ化学反应系统的稳定性和最终有界集估计。首先利用小增益定理给出该化学反应系统全局渐近稳定的充分条件,最后通过构造一个恰当的Lyapunov函数,我们获得了该化学反应系统最终有界集的一个估计。更多还原

【Abstract】 Switched systems are an important class of hybrid systems which consists of several continuous-time subsystems or discrete-time subsystems or continuous-time and discrete-time subsystems and a rule that orchestrates among them. Switched systems have broad applications and show complicated behaviours because of the interaction between the continuous dynamics and the switching signals. Hence switched systems attract a lot of authors’ much attention.The BZ reaction is currently one of the most arresting chemical experiments and objects of theoretic research in nonlinear chemistry. It has received much attention because the BZ reaction is the simplest reaction, and can show rich dynamical behaviours.In this dissertation we mainly study stability of switched systems with discrete-time subsytems. The main results are summarized as follows:1. For a general nonlinear nonautonomous discrete-time switched system, we first use multiple Lyapunov functions and the comparison principle of discrete systems to give a result on stability in terms of two measures. When the number of subsystems is finite, based on the result we present a detailed result on stability in terms of two measures. Subsequently we study several input-to-state stable(ISS)-type properties (input-to-state stable,1/λ-weighted input-to-state stable,1/λ-weighted sum input-tostate stable) of discrete-time switched systems with inputs, and we give some new results on several input-to-state stable(ISS)-type properties under average dwell-time switching. As an application of the result on input-to-state stability, sufficient conditions for the global asymptotic stability of cascade discrete-time switched systems are easily obtained.2. For a class of discrete-time switched systems-cascade switched systems, by means of the Brower fixed point theorem we first obtain a result on the existence of periodic orbits of this class of switched systems, and then we present a result on stability of periodic orbits. Furthermore we consider a particular case of this class of switched systems that this class of switched systems has one subsystem, and give some results on the number and the minimum period of peridic orbits. Some examples show that the results are unique for the particular case . 3. For a new type of switched systems-switched systems composed of continuous-time subsystems and discrete-time subsystems proposed in the literature, by combining the method of myultiple Lyapunov functions with the comparison principle of continuous-time systems and discrete-time systems we first present a result on stability in terms of two measures for such type of switched systems. When the numbers of continuous-time and discrete-time sunsystems are finite, on the basis of the above result we obtain a detailed result on stability in terms of two measures. Then we give a result on global asymptotic stability of such type of switched systems under average dwell-time switching. Moreover, for a special case of such type of switched systems-switched sytems composed of continuous-time LTI(linear time invariant)and discrete-time LTI subsystems, a result on global asymptotic stability of the switched systems under periodic switching signals is obtained.4. For switched systems composed of continuous-time subsystems and discrete-time subsystems, we discuss their robust stability. First we give a sufficient and necessary condition for quadratic stability of the linear case of the switched systems with polytopic pertubations, and then for a nonlinear case of the switched systems in which there are nonlinear pertubations in discrete-time subsystems we present an estimate of state under arbitrary switching laws, and from the estimate a sufficient condition for global asymptotic stability under arbitrary switching laws is easily obtained.Moreover, we investigate stability of the typical BZ reaction system and estimates ofthe ultimate bounded set, by means of the small gain theorem we first present sufficient conditions for global asymptotic stability of an equilibrium point of the chemical system . Then we give an estimate of the ultimate bounded set of the chemical system.更多还原